Lie group

A group that is also a manifold.
Lie groups were first described in the 19th century by the Norwegian mathematician
Sophus Lie (pronouned "Lee"). Lie
groups of real matrices, such as occur in
quantum field theory, give
naturally occurring examples of Lie groups. The tangent space at the identity
element of a Lie group forms a Lie algebra
in a natural way.

In layperson's terms, a Lie group is a way of describing symmetrical objects.
For example, there are Lie groups to describe the symmetry of simple objects
such as balls, cylinders, cones. The most complicated Lie group that exists
and, therefore, the most complicated symmetry known is a 248-dimensional
structure called E8.

E8

Part of the E8 matrix.
Credit: David Vogan, MIT

The Lie group known as E8 was discovered in 1887 but fully mapped
only in 2007 after several years of effort by a team of 18 mathematicians.
The final expression of it consisted of a 453,060 × 453,060 matrix.
Each of the 205,263,363,600 entries in this matrix consists of equations,
some of which are quite intricate.

E8 is much more than a mathematical curiosity, however. Physicists
have enoucntered it increasingly in their efforts to unify gravity with
the other fundamental forces in nature through theories such as string
theory. The mapping of E8 may thus play a significant role
in developing a consistent theory of quantum gravity.